Problem 54 - Algebra for College Students

Question

The expression \(0.06(10-x)(100)\) is equivalent to which one of the following? A. \(0.06-0.06 x\) B. \(60-6 x\) C. \(6-6 x\) D. \(6-0.06 x\)

Step-by-Step Solution

Answer

The expression is equivalent to \(60 - 6x\).

Step 1 - Distribute the constants

First, distribute the constant 0.06 to both terms inside the parentheses. This is done by multiplying 0.06 by each term separately: \(0.06(10-x)(100)\).

Step 2 - Multiply the first term

Multiply 0.06 by 10: \(0.06 \times 10 = 0.6\).

Step 3 - Multiply the second term

Multiply 0.06 by -x: \(0.06 \times (-x) = -0.06x\).

Step 4 - Multiply the result by 100

Now multiply the entire expression by 100: \(100 \times (0.6 - 0.06x)\).

Step 5 - Simplify

Distribute the 100 into the terms within the parentheses: \(100 \times 0.6 - 100 \times 0.06x\). This simplifies to \(60 - 6x\).

Key Concepts

distribution in algebra

Distribution in algebra is a process that helps us simplify expressions and solve equations. It involves multiplying each term inside a set of parentheses by a number outside the parentheses. For example, in the expression \(0.06(10 - x)(100)\), we can simplify it step-by-step using distribution.

First, we distribute the 0.06 to both terms inside the parentheses. This means we multiply 0.06 by 10 and 0.06 by \(-x\). This gives us two new terms: 0.6 and -0.06x. This process is based on the distributive law of multiplication over addition or subtraction, which states that \(a(b + c) = ab + ac\) or \(a(b - c) = ab - ac\).

Understanding distribution is crucial in algebra, as it allows us to break down more complex expressions into simpler parts.

multiplying constants

When we talk about multiplying constants in algebra, we're referring to the process of multiplying fixed numbers (constants) by variables or other constants. In the given problem \(0.06(10-x)(100)\), once we have distributed 0.06 inside the parentheses, we need to multiply 0.6 and -0.06x by another constant, which is 100.

This involves simple multiplication: \(100 \times 0.6 = 60\) and \(100 \times -0.06x = -6x\).

By understanding these multiplication steps, handling larger expressions becomes easier. Thus, multiplying constants accurately is an essential skill in solving algebraic expressions.

simplification

Simplification is the process of reducing an algebraic expression into its simplest form. After distributing and multiplying in the previous steps, the final step in the given problem is to simplify the expression.

We started with \(0.06(10 - x)(100)\) and through distribution and multiplication, we reached \(100 \times 0.6 - 100 \times -0.06x\).

Simplifying these terms gives us a final expression: \60 - 6x\. This process shows how complex expressions can be broken down systematically into simpler parts.

Simplification not only makes the problem easier to understand but also helps in accurately solving algebraic equations.